Automatic Classification Using Neural Networks - AU Journal

Gamal A. M. Al-Shawadfi, Hindi A. Al-Hindi
Automatic Classification Using Neural Networks
Gamal A. M. Al-Shawadfi and Hindi A. Al-Hindi
College of Business and Economics,
King Saud University,
Al-Qasseem Branch, Al-Melaida
Saudia Arabia
ABSTRACT
This paper proposes an artificial neural
network (ANN) to perform linear and
nonlinear classification of objects into several
classes. The theoretical and practical aspects
of the proposed approach are introduced, and
its validity was evaluated by the rate of correct
classifications. A Matlab macro program was
written for automatic classification using an
artificial neural network for linear and
nonlinear classification problems. The
network is designed, trained and tested with
different sample sizes. The results were
compared to those obtained from Fisher
discriminant function. In contrast with the
classical classification procedures, ANNs do
not require any pre assumptions about types of
data, distributions or the variance covariance
matrices. The numerical results illustrate the
capabilities of ANNs in solving linear and
nonlinear classification problems.
Keywords: Automatic Classification, Artificial
Neural Networks, Fisher Discriminant
Function.
1. INTRODUCTION
Classification is a multivariate technique
concerned with allocating new objects (or
observations) into previously defined groups
(populations). A distinction should be made
between 'classification' and ‘discriminant
analysis’. Discriminant analysis is a
multivariate technique concerned with
separating distinct sets of objects and often
employed on a one time basis in order to
investigate observed differences when causal
relationships are not well understood (Johnson
and Wichern, 1992). On the other hand,
classification is the problem in which an object
is assigned to one of several classes and
usually requires supervised learning methods
in which objects are assigned to known groups
(Fausett, 1994).
There are many studies related to
classification and neural network fields.
Wernecke et al. (1995) discussed the
validation of Classification Trees. Arminger
and Enache (1995) illustrated the relation
between statistical models and artificial neural
networks. More details on fundamentals of
neural networks, architectures, algorithms,
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Automatic Classification Using Neural Networks
applications and classification can be found in
Fausett (1994). The classical classification
approaches are found in some text books, see
for example, Johnson and Wichern (1992) and
Stevens (1992). Some neural network related
concepts and applications are found, for
example, in Elman (1993), Hertz et al. (1991),
Nerrand et al. (1993) and Tsoi and Tan (1997).
In recent years, the use of artificial neural
networks has increased for solving a wide
range of problems including pattern
recognition, classification and functional
approximation. An artificial neural network
consists of a number of simple processing
units or neurons and connections. Each simple
processing unit is associated with two major
functions: a summation function and an
activation function. The summation function
sums the inputs coming to the neuron from
other neurons or from the outside
environment. The activation function
determines the activation level of the neuron
based on the inputs it receives. Connections
represent the links between nodes and each
connection is associated with a weight that
reflects the strength of the relationship
between the connected nodes. Training the
network is the process of developing a
function that maps input vectors to output
vectors with minimum errors using a set of
training examples that includes input and
output data pairs.
For many practical cases, a feed forward
neural network with three different layers is
appropriate because it has been proven to be
an efficient system for representing nonlinear
relationships between a set of input and output
vectors to a high degree of accuracy. The first
layer is the input layer whose nodes receive
the inputs from outside. The second layer is
the hidden layer whose nodes receive inputs
from the input neurons and propagates them to
the output nodes. The third layer is the output
layer whose nodes determine the outputs of the
neural network.
In this paper, we propose a neural network
approach to perform automatic classification
and compare its performance with the classical
classification method. With the classical
classification method, all groups are assumed
to have the same variance-covariance matrix
and the distributions are normal. In the
absence of these two assumptions, linear or
quadratic classification rules are inadequate
(Johnson and Wichern, 1992). The neural
network classification technique does not
impose such assumptions.
The performance of the two classification
techniques is illustrated with three data sets
that use two variables in classifying objects
into three categories. The first data set
describes the case of linear classification. The
second and third describe the case of nonlinear
classification with varying degrees of
nonlinearity.
The rest of the paper is organized as
follows. The second section presents the
proposed automatic classification method. The
third section presents the simulated
classification problems to test the proposed
approach. The fourth section summarizes this
paper.
2. PROPOSED ANN AUTOMATIC
CLASSIFICATION METHOD
Let us start with a description of the kind
of data for which the proposed classification
method is appropriate. The data consists of
objects, and their corresponding descriptions.
The objects may be quantitative, for example
students cumulative rates, or qualitative, for
example documents, keywords, hand written
characters and species. Qualitative data may
be replaced with quantitative data according to
suitable coding scheme. Then the
classification method is used to summarize and
simplify the data.
International Journal of The Computer, The Internet and Management, Vol. 11, No.3, 2003, pp. 72 - 82
73

Gamal A. M. Al-Shawadfi, Hindi A. Al-Hindi
Classification rules are developed from
training samples. Usually, the data are
separated into two subsets, where the first,
called training set, is used to fit a suitable
discriminant function, and the second, called
the test set, is used to check the validity of the
discriminant function for classification. In the
ANN model, the data need to be normalized
usually into the range [0,1] before training. In
this research, the following equation is used to
normalize the data:
? y - y ?
x = 0.8
max i
+ 0.1 (1)
i ? y - y ?
? max min ?
where y max and y min are the respective
maximum and minimum values of all
observations.
Training in neural networks is the process
of adjusting the weights of the network
connections in order to obtain the desired
outputs (z i ) given a set of inputs (x i1 , x i2 , … ,
x im ). The discriminant function is:
z = f
) +
i
i( xi
1
, xi
2,
. . . . ., xim
ui
(2)
where z i represents the classes (outputs),
i=1,2,…..,k and fi( xi
1,
xi
2,
. . . . ., xim)
is a
linear or nonlinear function in the input
variables x
i
, x , 1 i 2
. . . . ., x im
and ui represents
errors between the input and the output values.
Equation (2) represents a class of artificial
neural networks which may be interpreted as a
complex multivariate statistical model for the
approximation of an unknown expectation
function of a random variable z given an
explanatory variables x i ’s, i=1,2,….,m This
class of models represents a discriminant
function.
The training process includes two stages:
forward and backward pass. In the forward
pass, the input neurons receive input examples
and pass them to hidden layer. Each neuron
calculates its activation level using the
summation function to sum the weighted
inputs. This sum is then used by the activation
function to determine the output of the neuron.
Usually, the activation function is a linear
combinations of parameters and inputs. In
practice, many types of transfer functions may
be used, which include: linear combination,
normal distribution, logistic distribution,
hyperbolic tangent, indicator function and
threshold functions. The selected activation
function here is a sigmoidal function. Sigmoid
activation functions are easy to differentiate
and make computation of the gradient easy.
The sigmoid function has the form:
1
f
j
( x 1
,..., x n
) =
n
(3)
? w ij + b
1+
j
i = 1
e
where f j
( x 1
,..., x n
) is the output, x is the input
observations, w called weights, and b is the
bias term. The values of w and b should be
initialized before the training phase. In Matlab
package, the function INITFF is used to
determine initial values for w and b.
The outputs of the hidden neurons are
used as input to neurons in the output layer.
Neurons in the output layer use the summation
and activation functions to determine the
outputs of the neural network. The generated
outputs is then compared to the actual outputs
and the resulting errors is propagated back
through the network in the backward pass of
the training algorithm.
The weights w jk that connect neurons in
the hidden layer with neurons in the output
layer are updated first. Then, the weights w ij
that connect neurons in the input layer with
neurons in the hidden layer are updated. There
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Automatic Classification Using Neural Networks
are two strategies that can be used to update
the connection weights. The first is to
calculate the error for each training example
and propagate this error back to update the
weights. The second strategy is to calculate the
errors for all training examples and use this
sum to update the weights. The iterative
process of the back propagation training
algorithm continues until the error function
reaches a predetermined level (say 0.001), or
the number of iterations is satisfied. The error
function is:
1
2
MSE = ? ( c
j
− c
j
( NNC))
(4)
n
where MSE is the mean of sum of squares of
the difference between the actual output (c j )
and NN estimated output c j (NNC).
Due to their capabilities, neural networks
may be used to approximate continuous
mapping functions. The following theorem of
Kolmogorov shows the existence of the
mapping neural network.
Theorem: Any continuous function
f ( xi ), i = 1,2, . . n of several variables defined
on I
n , where n ≥ 2 and I =[0,1] , can be
represented in the form :
2n
? + 1
j
j = 1
f ( x)
= b w x (5)
ij
i
where b j and w ij are continuous functions of
one variable and w ij are monotonic functions
that do not depend on f.
The theorem states that: “a feed forward
neural network with three layers of neurons
(input, hidden and output units) can represent
any continuous function exactly”, see Fausett
(1994) pp.328-329.
A good classification procedure should
result in few misclassifications. The validity of
the proposed NN classification method may be
checked using the apparent error rate (APER).
The apparent error rate is the fraction of
observations in the test set that are
misclassified by the discriminant function. The
apparent error rate is calculated according to
the equation:
APER
k
?
i=
= k
?
i=
1
ni
1
(6)
N
i
where N i is the number of observations in
group i, i = 1,2,….,k , and n i is the number of
items misclassified in another group other
than i. However, we can measure the validity
of the neural network classification procedure
by calculating the rate of correct classification
from the following equation:
CCR
k
?
ni
1 (7)
N
i=
1
= −
k
?
i=
1
i
The correct classification rate (CCR)
represents the item in the test set that are
correctly classified. This measure does not
depend on the form of the parent population,
also it can be calculated for any classification
procedure.
To do automatic classification for data in
several classes, we designed a Matlab macro
program presented in the appendix. In the
program we use essentially three functions
from neural network toolbox. INITFF to
initialize feed-forward network up to three
layers. TRAINBPX to train a feed-forward
network with fast back propagation and can be
invoked with 1, 2, or 3 sets of weights.
SIMUFF to simulate a feed-forward network
with up to 3 layers.
International Journal of The Computer, The Internet and Management, Vol. 11, No.3, 2003, pp. 72 - 82
75

Gamal A. M. Al-Shawadfi, Hindi A. Al-Hindi
In summary, the steps of the proposed NN
classification approach are:
1. Designing the network: Use equation 1 to
scale the data between 0, 1 interval, select
a suitable discriminant function, determine
the number of training epochs, and error
level, and select initial values for the
weights and biases.
2. Training the network: Spilt the data into
two groups, and use the first group to train
the network and estimate the parameters
w i , b i of the discriminant function.
3. Checking the validity: Check the validity of
the proposed discriminant function, using
the second group of data with equation (7)
which gives the rate of the correct
classifications.
4. Classifying new observations: If step 3 is
satisfied, use the proposed discriminant
function in classifying new observations
into suitable groups, otherwise repeat the
training process with other function and
weights to obtain a more efficient
discriminant function.
The following sets of examples are used to
compare the performance of neural networks
with classical classification.
3. SOME EXAMPLES
Three different groups, with different
properties regarding the relationships between
variables, are used in the comparative
procedure. The first group shows the
classification in case of linear relationship
between inputs and outputs. The second and
the third groups show the classification in the
case of nonlinear relationships between the
inputs and the outputs. In all examples, both
the proposed neural network and Fisher
classification approaches are used in solving
the problem of classifying objects into three
categories according to some inputs. The
inputs are two variables related linearly or
nonlinearly to the output variable. For each
group, different sample sizes are used to show
the effect of increasing the sample size on
classification accuracy. The selected samples
sizes are 10, 30, 50, 70, 90, 110, 130 and 150.
The validity of the classification results are
tested by the rate of correct classifications
(CCR).
The four steps of the proposed neural
network classification mentioned in section (2)
were applied here: designing the network,
training the network, checking the validity,
and classifying new observations.
3.1 The Case of Linear Classification
In this case, the output variable (or object)
is classified into three categories according to
two variables (inputs). The data samples are
generated according to the following linear
equation:
y 3 = 0.5 y 1 + .5 y 2 (8)
where y 1 and y 2 are input variables and y 3 is
the output. Our aim is to use the proposed NN
procedure to find a function to allocate an
observation from y 3 into class 1, 2 or 3
according to the values of y 1 and y 2 .
In Fisher classification procedure, sample
discriminants are used to allocate x i into
population k if:
r
?
j = 1
r
?
j = 1
( y
j
− y
=
( λ ( xˆ
− x ))
j
kj
)
i
2
2
r
?
j = 1
( λ ( xˆ
− x ))
j
k
2
?
(9)
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Automatic Classification Using Neural Networks
For all i ? k , where λj is the eigenvectors of
w -1 B 0 and r ? s .
Table 1:
The NN and classical classifications of
objects into 3 classes
y 3 = 0.5 y 1 + 0.5 y 2
N NN CCR F. CCR
10 0.9000 0.8000
30 0.8333 0.6667
50 0.7400 0.7600
70 0.9000 0.7571
90 0.7778 0.9333
110 0.9727 1.0000
130 0.8385 0.9923
150 0.8600 0.9667
average 0.8528 0.8595
Table 1 shows the rate of correct
classification (CCR) for each method. The first
column represents the sample size. The second
column shows the rate of correct classification
(CCR) for the neural network approach.
Column three shows the rate for the Fisher
classification method. As it is shown in the
table, the rates of correct classification for the
two methods are very close in average, the
average of correct classification for the neural
network approach is 85.29% and 85.95% for
the classical classification approach.
3.2 The Case of Nonlinear Classification
For this case, data samples were generated
to show the capabilities of the proposed neural
network approach in solving nonlinear
classification problems. Each new observation
is classified into one of three categories
according to two input variables. The data
were generated according to the equation:
y 3 = 0.5 y 1 + 0.5 y 2
2
(10)
Table 2 shows the NN and Fisher
classifications of objects into 3 classes for the
nonlinear case. We find that the overall
performance of the neural network nonlinear
classification approach is fairly very good, in
the sense that the average rate of correct
classifications is 84.74%. With respect to the
sample sizes, these rates fluctuate between
63.33% and 100%. On the other hand, the
rates of Fisher classification approach
fluctuate between 60% and 84.29%.
Table 2:
The NN and classical classification of
objects into 3 classes
y 3 = .5y 1 + .5y 2
2
N
NN CCR F. CCR
10 1.0000 0.7000
30 0.6333 0.6000
50 0.8400 0.8200
70 0.7000 0.8429
90 0.9556 0.7556
110 0.9200 0.7200
130 0.8700 0.8300
150 0.8600 0.8100
average 0.8474 0.7598
The other set of data samples for the nonlinear
classification was generated according to the
following equation:
y 3 = 0.5 y 1 + 0.5 y 2
3
(11)
From table 3, we find that the performance of
the NN nonlinear classification approach is
extremely high where the average of correct
classification rate is 97.5%. The rates fluctuate
between 80% and 100%. On the other hand,
the rates of the classical classification
approach fluctuate between 72.86% and
96.67%. Also, the average rate is only 81.4%.
International Journal of The Computer, The Internet and Management, Vol. 11, No.3, 2003, pp. 72 - 82
77

Gamal A. M. Al-Shawadfi, Hindi A. Al-Hindi
Table 3:
The NN and classical classifications of
objects into 3 classes
y 3 = .5y 1 + .5y 2
3
N NN CCR F. CCR
10 0.8000 0.8000
30 1.0000 0.9667
50 1.0000 0.8200
70 1.0000 0.7286
90 1.0000 0.7667
110 1.0000 0.7600
130 1.0000 0.8200
150 1.0000 0.8500
average 0.9750 0.8140
complexity of the discriminant function
depends on the number of layers as well as the
number of units in the hidden layer (hidden
units). Another advantage is that the method
does not put any constraints on population
distribution or require the equivalence of the
population variance-covariance matrices. The
numerical examples support the suitability of
the proposed approach in data classification.
The findings of the study draw several
distinctions between neural network and
classical classification: i) for linear
classification, the performance of the two
methods is equivalent, ii) for nonlinear
classification, neural networks outperform the
classical classification method and iii) the
relative performance of the neural network
increases as the level of nonlinearity increases.
4 CONCLUSION
This paper proposes a neural network
method to do nonparametric automatic
classification. The proposed method may be
used in classifying a sample of n objects into k
classes according to linear or nonlinear
discriminant function. The discriminant
function is obtained from a training process of
the neural network. The selected discriminant
function is the one which minimizes the
difference between the actual and the expected
numbers of classes. The validity of the
proposed discriminant function is measured by
the rate of correct classifications (CCR), where
the discriminant function is acceptable if CCR
is more than some ratio (say 50%).
Classification using NN has several
advantages compared to the classical methods.
One advantage is the possibility of
constructing nonlinear discriminant function
for solving sophisticated classification
problems. Using the proposed approach, a very
complex nonlinear approximation functions
can be built from simple components. The
REFERENCES
Arminger, G. and D. Enache (1995),
“Statistical Models and Artificial Neural
Networks”, In Proceedings of the 19th
annual conference of the Gesellschaft fur
classification e. V., University of Basel,
H.-H. Bock and W. Polasek Editors ,
Springers , Germany.
Cox, Hinkly (1979), Theoretical Statistics,
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David, K. Hildebrand and Lyman oat (1991)
Statistical Thinking for Managers, Fourth
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Multivariate Analysis in SAR and
Environmental Studies, Kluwer academic
Publishers: London , U.K.
Elman, J. L. (1993), “Learning and
Development in Neural Networks: the
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Automatic Classification Using Neural Networks
Importance of Starting Small”, Cognition,
48, pp.71-99.
Fausett, Laurene (1994), Fundamentals of
neural networks, Architectures,
algorithms And Applications, Prentice
Hall Inc.: New York.
Hertz J. and et al. (1991), Introduction to the
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Modern Approach to Statistics, John
Wiley & sons: New York, U.S.A.
Johnson, Richard A. and Dean, W. Wichern
(1992), Applied Multivariate Statistical
Analysis, Third Edition, Prentice Hall,
Englewood Cliffs: New Jersey, U.S.A.
Matlab Reference Manual (1997), Version 5.3,
Mathworks, U.S.A.
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Stevens, James (1992), Applied Multivariate
Statistics for the Social, Sciences, Second
Edition, Lawrence, Erlbaum Associates
Publishers: Hillsdale, New Jersey, U.S.A.
Tsoi A.C. and Tan S. (1997), “Recurrent
neural networks: A constructive algorithm
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pp.309-326.
Wernecke, K. D.; Possinger, K. and Kalb,
G. (1995), “On the Validation of
Classification Trees”, Proceedings of the
19th Annual Conference of the
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University of Basel, March 8-10 , 1995 ,
Bock and Polasek Editors, Springer:
U.S.A.
Zar, Jerrold H. (1984), Bio-statistical Analysis,
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Naftaly, U.; Intrator, N. and Horn, D., (1997),
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Nerrand, O.; Roussel-Ragot, P.; Personnaz, L.;
Dreyfus, G. and Marcos, S. (1993),
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International Journal of The Computer, The Internet and Management, Vol. 11, No.3, 2003, pp. 72 - 82
79

Gamal A. M. Al-Shawadfi, Hindi A. Al-Hindi
APPENDIX
% ... Matlab Macro Program for Automatic Classification of Data ...
%... ...file name : train1394 ... output file oo1394.mat... ...
disp '... ...Examples of NN and Fisher Classification'
disp '... ...for Data Generated From Binomial Distribution '
diary ('oo1394')
clc;clear all;an=320;mm=an/40;aa(mm,4)=0;nn=15;
r1=binornd(nn,.5,an,1);r2=binornd(nn,.3,an,1);r3=.5*r1+.5*(r2);
for j =1:mm
j
n = 20+40*(j-1);
n1=0;n2=0;n3=0;nn1=0;nn2=0;nn3=0;
c1 = round(n/2);
if n< 100
c2=n-c1 ;
else c2=50;
end
m = 5;k =3 ;d(k)=0;
ss(k,1)= 0 ; ss0(k,1) = 0 ; ss1(k,1) = 0 ;st0(4)=0 ;st(k)=0;st1(4)=0;
x1(k)=0;x2(k)=0;y1(c1)=0;y2(c1)=0;z3(c1) = 0 ;
r03=r3(1:c1) ; z1=r1(1:c1) ; z2=r2(1:c1) ;
%1... ... ... ... INITIALIZATAION OF DATA ... ... ... ...
r =(max(r03) - min(r03))/3;
r0 = min(r03)+r;
for i = 1 : c1
if r03(i) < r0
z3(i) = 1 ;
n1 = n1+1 ;
x1(1)= x1(1)+z1(i);x2(1)=x2(1)+z2(i);
elseif r3(i) > r0+r;
z3(i)=3 ;
n3=n3+1;
x1(3)=x1(3)+z1(i);x2(3)=x2(3)+z2(i);
else
z3(i)= 2;
n2=n2+1;
x1(2)=x1(2)+z1(i);x2(2)=x2(2)+z2(i);
end
end
% ... Fisher method for classification ... ... ...
x01= ones(1,c1)*z1/c1;x02=ones(1,c1)*z2/c1;
x11= x1(1)/n1 ; x21=x1(2)/n2 ; x31=x1(3)/n3;
x12= x2(1)/n1 ; x22=x2(2)/n2 ; x32=x2(3)/n3;
xxx= [x11 x12;x21 x22; x31 x32];
xx1= [x11-x01;x21-x01;x31-x01];
xx2= [x12-x02;x22-x02;x32-x02];
for i = 1 : c1
if r03(i) < r0
y1(i)= z1(i)-x11;
y2(i)= z2(i)-x12;
elseif r3(i) > r0+r;
y1(i)= z1(i)-x31;
y2(i)= z2(i)-x32;
else
y1(i)= z1(i)-x21;
y2(i)= z2(i)-x22;
end
end
x =[ xx1 xx2] ;
80

Automatic Classification Using Neural Networks
B0= transpose(x)*x;
y=[y1 ; y2] ;
yy= y*transpose(y);
ww=inv(yy);
ww1=ww*B0;
[cv a] = eig(ww1);
v01=cv(:,1);v02=cv(:,2);
cc1=transpose(v01)*yy*v01;
cc2=transpose(v02)*yy*v02;
c01=sqrt((n-3)/cc1);
c02=sqrt((n-3)/cc2);
cv1=v01*c01;cv2=v02*c02;
v=[cv1 cv2];
% ... Proposed NN method for classification ... ... ...
z3=transpose(z3);
z=[z1 z2 z3];
z0 = (0.8*(z-ones(c1,1)*min(z))./(ones(c1,1)*(max(z)-min(z))))+0.1;
z01 = transpose(z0(:,1:2)); z02 = transpose(z0(:,3));
k1='purelin' ; k2='logsig' ; k3='tansig';
[w1,b1,w2,b2]=initff(z01,m,k2,z02,k2) ;
[w1 , b1 ,w2, b2,TE,TR]=trainbpx(w1,b1,k2,w2,b2,k2,z01,z02,[50,5000
,.001 ,.001]);
%2... ... ... ...comparison and testing phase ... ... ... ...
zz1=r1((c1+1):c1+c2,:); zz2=r2((c1+1):c1+c2,:); rr03=r3((c1+1):c1+c2,:);
zzz=[zz1 zz2];
v1 = zzz*v;
v2 = xxx*v;
%1... ... ... ... INITIALIZATAION OF DATA ... ... ... ...
rr =(max(rr03) - min(rr03))/3;
rr0=min(rr03)+rr;
for i = 1:c2
if rr03(i)< rr0
zz3(i) =1;nn1=nn1+1;
elseif rr03(i) > rr0+rr
zz3(i)=3;nn3=nn3+1;
else
zz3(i)= 2; nn2=nn2+1;
end
end
zz3=transpose(zz3);
zz=[zz1 zz2 zz3];
zz0 = (0.8*(zz-ones(c2,1)*min(zz))./(ones(c2,1)*(max(zz)-
min(zz))))+0.1;
zz01 = transpose(zz0(:,1:2));
zz02 = transpose(zz0(:,3));
for i=1:c2
d(1)= (v1(i,:)-v2(1,:))*transpose((v1(i,:)-v2(1,:)));
d(2)= (v1(i,:)-v2(2,:))*transpose((v1(i,:)-v2(2,:)));
d(3)= (v1(i,:)-v2(3,:))*transpose((v1(i,:)-v2(3,:)));
[d1,dd1] = min(d);
yf(i)=dd1;
if dd1 < 1.5 & zz3(i)==1 ;st1(1)=st1(1)+1;
elseif dd1>= 1.5&dd1 < 2.5 & zz3(i)==2 ;st1(2)=st1(2)+1;
elseif dd1>=2.5
& zz3(i)==3 ;st1(3)=st1(3)+1;
else
;st1(4)=st1(4)+1;
end
s = simuff(zz01(:,i),w1,b1,k2,w2,b2,k2);
s1= (((s- 0.1)*(max(zz3)-min(zz3)))/0.8 )+ min(zz3);
y0(i)=s1;
if s1 < 1.5 & zz3(i)==1 ;st0(1)=st0(1)+1;
elseif s1 >= 1.5&s1 < 2.5 & zz3(i)==2 ;st0(2)=st0(2)+1;
elseif s1 >=2.5 & zz3(i)==3 ;st0(3)=st0(3)+1;
International Journal of The Computer, The Internet and Management, Vol. 11, No.3, 2003, pp. 72 - 82
81

Gamal A. M. Al-Shawadfi, Hindi A. Al-Hindi
else
;st0(4)=st0(4)+1;
end
end
ss1 = (st0(1)+st0(2)+st0(3))/c2;
ss2 = (st1(1)+st1(2)+st1(3))/c2;
%3...Results ... ...
aa(j,:)=[j c1 ss1 ss2];
% [zz3 transpose(yf) transpose(y0)]
clear b* ; clear B* ; clear c* ;clear d* ; clear n* ;
clear s* ;clear v* ; clear w* ;clear x* ; clear y* ;clear z*;
end
disp ' HIT RATE RESULTS Fisher class. & Neural class.'
aa a1= mean(aa)
save o1394;
diary off
82